Sep 28, 2016 - It is not an easy task to obtain probabilities for emptiness and ..... [1] Moran, P.A.P. (1954) A Probability Theory of Dams and Storage Systems.
American Journal of Operations Research, 2016, 6, 414-423 http://www.scirp.org/journal/ajor ISSN Online: 2160-8849 ISSN Print: 2160-8830
Stochastic Reservoir Systems with Different Assumptions for Storage Losses Carter Browning1, Hillel Kumin2 Halliburton Drill Bits and Services, Oklahoma City, OK, USA School of Industrial and Systems Engineering, University of Oklahoma, Norman, OK, USA
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How to cite this paper: Browning, C. and Kumin, H. (2016) Stochastic Reservoir Systems with Different Assumptions for Storage Losses. American Journal of Operations Research, 6, 414-423. http://dx.doi.org/10.4236/ajor.2016.65038 Received: August 15, 2016 Accepted: September 25, 2016 Published: September 28, 2016 Copyright © 2016 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/ Open Access
Abstract Moran considered a dam whose inflow in a given interval of time is a continuous random variable. He then developed integral equations for the probabilities of emptiness and overflow. These equations are difficult to solve numerically; thus, approximations have been proposed that discretize the input. In this paper, extensions are considered for storage systems with different assumptions for storage losses. We also develop discrete approximations for the probabilities of emptiness and overflow.
Keywords Stochastic Storage Systems, Storage Losses, Probability of Emptiness and Overflow
1. Introduction Moran [1] [2], Prabhu [3] [4] and Ghosal [5] all considered a finite dam whose input in a given interval of time is a continuous random variable. Integral equations are then developed that give the probability of emptiness and overflow. It is difficult to obtain exact numerical results from these equations. An analytic solution has only been obtained for an Erlang input. Klemes [6], Lochert and Phatarfod [7], Phatarfod and Srikanthan [8] and others have obtained approximations for these probabilities by discretizing the input. Following Bae and Devine [9], we consider reservoir systems with different assumptions for storage losses. We then obtain integral equations as above for the probability of emptiness and overflow, and develop discrete approximations to obtain numerical results for the probabilities of overflow and emptiness. Moran considered a storage model of a dam in discrete time, t = 0,1, 2, . Let Z t be the level of the dam before input X t , where the X’s are i.i.d. random variables. Let
DOI: 10.4236/ajor.2016.65038 September 28, 2016
C. Browning, H. Kumin
Yt be the release at the end of the time period ( t , t + 1) , where the Y’s are i.i.d. random variables independent of the X’s, and let k < ∞ be the capacity of the system. If Z t + X t > k , then there is an overflow of X t + Z t − k . If Z t + X t ≤ k then no overflow occurs. At the end of the period, if there is an overflow, then Z t +1= k − Yt . If there is no overflow, then either Z t +1 = Z t + X t − Yt or Z t +1 = 0 if the storage system is empty. Lindley [10] showed that if certain independence conditions are satisfied then
{
}
Ft +1 ( y ) Pr storage level Z t +1 of ( t + 1) period ≤ y = st
=Pr {Z t +1 =0} + Pr {0 < Z t +1 ≤ y} = Pr {Z t + X t − Yt ≤ 0} + Pr {0 < Z t + X t − Yt ≤ y} = Pr {Z t + X t − Yt ≤ y} .
where y ∈ [ 0, k ] . Further, define H ( ⋅) to be the c.d.f. of U t where
U= X t − Yt t Then, by convolution Ft= +1 ( y )
y
∫ Ft ( y − x ) dH ( x ) ,
0≤ y
